We begin with a description of the PCL setting. We observe a treatment \(A\in \mathcal {A}\) , where \(\mathcal {A} \subset \mathbb {R}^{d_A}\) , and the corresponding outcome \(Y\in \mathcal {Y}\) , where \(\mathcal {Y} \subset \mathbb {R}\) .
We assume that there exists an unobserved confounder \(U \in \mathcal {U}\)  that affects both \(A\)  and \(Y\) . The goal of PCL is to estimate the structural function \(f_\mathrm {struct}(X)\)  defined as

fstruct(a) = [U][Y]Y|A=a, U,

which we assume to be continuous. This function is also known as the Average Treatment Effect (ATE).
The challenge of estimating \(f_\mathrm {struct}\)  is that the confounder \(U\)  is not observable — we cannot estimate the structural function from observations \(A\)  and \(Y\)  alone. To deal with this, we assume access to a treatment-inducing proxy variable \(Z\) , and an outcome-inducing proxy variable \(W\) , which satisfy the following structural assumption and completeness assumption.
[Structural Assumption [1]}, [2]}] 
We assume \(Y Z | A, U\) , and \(W (A,Z) | U\) .

[Completeness Assumption on Confounder [1]}, [2]}] 
Let \(l: \mathcal {U} \rightarrow \mathbb {R}\)  be any square integrable function \(\Vert l\Vert _{{U}} < \infty \) .
The following conditions hold:

l(U) A=a, W=w=0   (a,w) A W       l(u) = 0    U-a.e.
